Transcript Bridges Baltimore, July 2015 Large, “7-Around” Hyperbolic Disks Sean Jeng Liu, Young Kim, Raymond Shiau, Carlo H.

Bridges Baltimore, July 2015
Large, “7-Around” Hyperbolic Disks
Sean Jeng Liu, Young Kim,
Raymond Shiau, Carlo H. Séquin
University of California, Berkeley
Assembling Equilateral Triangles
5 per vertex:
6 per vertex:
7 per vertex:
pos. curved:
 Icosahedron
flat:
 a plane
neg. curved:
 hyperbolic
Hyperbolic Surface: Poincaré Disk Model

Scaling allows to accommodate infinitely many triangles.
How much of that infinite tiling
can we accommodate with
equilateral triangles ?
Amy Ione & CW Tyler
David Richter
Better Luck with Soft Materials
Gabriele Meyer
(posted by Loren Serfass)
Computer
Model by
Mark Howison
(2007)
Best result:
810 Triangles
(20 hrs CPU time)
Extending the Disk . . .
 By
adding
full annuli –
one at a time,
with ever more triangles …
 Exploit:
6-fold
D3-symmetry
New Approach: Add 6-fold Symmetry
810
2197
Howison’s annuli:
–
starting with a central vertex.
Our new annuli:
starting with a central triangle.
Starting with a Symmetrical Core

D3 symmetry
forces some
constraints:

The 4 central
triangles are
coplanar!

Yellow-olive
edges lie on
symm. axes;
adj. triangles
are coplanar.
Constructing an Extended Core

Blue-teal
edges lie on
symm. axes.
The two
triangles are
coplanar!

Give #4 & #2
the same sign
for the z-value
to make
“nice, looping
arch”
Manually Constructed, Extended Core
coplanar
 61 triangles with D3 symmetry with nice undulating border.
Step-by-Step Construction

Complete one vertex at a time: “3”, “4”, “5”, “6”, “7”, “8”
in orange “swath #1” throughout a 60 sector.
e.g. vertex “5”:
add “a”, “b”, “c” ind.;
last two as a “gable.”
a b
5
given
c
Interference Checking & Back-Tracking


Interference:

Check for intersections between triangles;

Apply conservative proximity check for inner swath;

If fails, we try different dihedral angles:

Randomly; or

Direction-specific (as determined by program); or

Manual tweaking upon visual inspection
Back-track if:

Fails intersection checks after multiple tries; or

Fails to meet other heuristic guidelines
Overall Strategy: 1 Annulus at a Time

We only have to construct one 60 sector of the
whole disk, which then gets replicated 6 times.

We construct this one “swath” (= 1/6 annulus)
at a time; we try to construct a “good” swath,
one that leaves most space for subsequent one.

Such a good swath gets added to the “core”;
it now acts as a starting point for the next swath.
Results
May 2015
July 2015
Video
Fully Instantiated Disk (May 2015)
Final Result at Conference Time
2197 triangles
Conclusions

Computers are useful and powerful;
but brute-force approaches may only get limited results.

Use your brain to gain an understanding of the problem,
and the tailor your search to make use of such insights.

A good combination of the two approaches
can then result in a more effective search,
reducing computation time exponentially.

This is often true in engineering problems relying on
simulated annealing or on genetic algorithms.